Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Cockayne, Jon; Oates, Chris J.; Sullivan, Tim; Girolami, Mark

doi:10.1063/1.4985359

Cited by 79 publications

(85 citation statements)

References 56 publications

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“…Remark 3. Similar to our approach, Probabilistic Numerical Methods (PNMs) [63,64,65] take a statistical point of view of classical numerical methods (e.g. a finite element solver) that treat the output as a point estimate of the true solution.…”

Section: Probabilistic Surrogates With Reverse Kl Formulationmentioning

confidence: 99%

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Zhu

Zabaras

Koutsourelakis

et al. 2019

Journal of Computational Physics

761

458

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Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.

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Section: Probabilistic Surrogates With Reverse Kl Formulationmentioning

confidence: 99%

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Zhu

Zabaras

Koutsourelakis

et al. 2019

Journal of Computational Physics

761

458

View full text Add to dashboard Cite

show abstract

“…For treatment of PDEs possible choices of index variables in (4) or (6) include separation constants of analytical solutions, or the frequency variable of an integral transform. In accordance with [10], using basis functions that satisfy the underlying PDE, a probabilistic meshless method (PMM) is constructed. In particular, if ζ ζ ζ parameterizes positions of sources, and φ (x, ζ ζ ζ ) = G(x, ζ ζ ζ ) in (6) is chosen to be a fundamental solution / Green's function G(x, ζ ζ ζ ) of the PDE, one may call the resulting scheme a probabilistic method of fundamental solutions (pMFS).…”

Section: Construction Of Kernels For Pdesmentioning

confidence: 99%

“…In particular, if ζ ζ ζ parameterizes positions of sources, and φ (x, ζ ζ ζ ) = G(x, ζ ζ ζ ) in (6) is chosen to be a fundamental solution / Green's function G(x, ζ ζ ζ ) of the PDE, one may call the resulting scheme a probabilistic method of fundamental solutions (pMFS). In [10] sources are placed across the whole computational domain, and the resulting kernel is called natural. Here we will instead place sources in the exterior to fulfill the homogeneous interior problem, as in the classical MFS [6,7,8].…”

Section: Construction Of Kernels For Pdesmentioning

confidence: 99%

“…This work also provides a detailed treatment of kernels for the 2D Laplace equation. A more extensive and general treatment of the Bayesian context as well as kernels and their connection to fundamental solutions is available in [10] under the term probabilistic meshless methods (PMM).…”

Section: Introductionmentioning

confidence: 99%

“…While [9] is focused on boundary data of a single homogeneous equation, and [10] provides a detailed mathematical foundation, the present work aims to explore the topic further for application and extend the recent work in [11]. Starting from general notions some regression techniques are introduced with emphasis on the role of localized sources.…”

Section: Introductionmentioning

confidence: 99%

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Gaussian Processes for Data Fulfilling Linear Differential Equations

Albert

2019

The 39th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering

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A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The approach is applicable to a wide range of data from physical measurements and numerical simulations. It is based on the well-known invariance of the Gaussian under linear operators, in particular differentiation. Instead of using a generic covariance function to represent data from an unknown field, the space of possible covariance functions is restricted to allow only Gaussian random fields that fulfill the homogeneous differential equation. The resulting tailored kernel functions lead to more reliable regression compared to using a generic kernel and makes some hyperparameters directly interpretable. For differential equations representing laws of physics such a choice limits realizations of random fields to physically possible solutions. Source terms are added by superposition and their strength estimated in a probabilistic fashion, together with possibly unknown hyperparameters with physical meaning in the differential operator.

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A strong form based moving Kriging collocation method for the numerical solution of partial differential equations with mixed boundary conditions

Bott

Gee

2020

Numerical Meth Engineering

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In this article, a novel mesh‐free, moving Kriging (MK) based collocation scheme for the numerical solution of partial differential equations (PDEs) is introduced. In contrast to methods that are based on a Galerkin weak form of the governing PDEs, the MK collocation (MKC) approach, which is strong form based, is truly mesh‐free in the sense that no background mesh is required for numerical integration. In fact, the presented approach does not require the evaluation of any integrals. Since the approximation function in the MK framework can be conditioned on point value‐ as well as derivative‐information, the pointwise exact imposition of essential as well as natural boundary conditions is rendered straightforward. By incorporating an explicit linear basis into the MK framework, the first‐order consistency condition is fulfilled, and thus rigid body motions are captured accurately. Moreover, Kriging functions may be conceived that comply with constraints on higher order derivatives such as the PDE at hand at certain locations. This possibility proves useful in improving the solution accuracy in the vicinity of Dirichlet boundaries. This article provides a study of the method's characteristics by means of 2D linear elasticity examples. It concludes with a suggestion on how to apply MKC to nonlinear PDEs.

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Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Cited by 79 publications

References 56 publications

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Gaussian Processes for Data Fulfilling Linear Differential Equations

A strong form based moving Kriging collocation method for the numerical solution of partial differential equations with mixed boundary conditions

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